Yes and no, it depends on what kind of fossils you've found and where you found them.

Vertebrate Fossils: Any vertebrate fossil (one that looks like a fossilized bone) found on federal, state, or tribal land cannot be legally collected without special permission. This protection is to ensure that these rare vertebrate fossils can be carefully excavated and studied by experienced paleontologists at research institutions, and can add to our understanding of our natural history heritage. Vertebrate fossils found on private land belong to the land owner, so permission must be obtained to collect from them as well. If you are determined to be part of a dinosaur excavation in New Mexico, you're best option is to contact one of the paleontology museums in New Mexico that runs a volunteer program.

Invertebrate Fossils:Plants and invertebrate animals (that don't have bones, like snails) are far more numerous that vertebrate fossils and are not accorded the same special legal protection on public lands. If you can collect mineral specimens, you generally can collect fossils as well on public lands. Note that many public lands, like parks, monuments, and wilderness areas RESTRICT ALL COLLECTION of natural resources. Please be aware of the rules that govern any particular spot before you collect.

I think that I have found a dinosaur bone. What should I do?

The best option is to leave the fossil where you discovered it, take photographs, be sure you can locate the site again (perhaps mark a spot on a map or use a GPS unit), and then contact one of the paleontology museums or paleontologists at one of the universities in New Mexico. Many of the best dinosaur finds are reported by the public, and the staff at these institutions are happy to respond to you.